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arXiv:2607.03274

False vacuum decay in long-range interacting quantum systems

Valerio Pagni, Laura Batini, Nicolò Defenu

quant-phgr-qchep-thSubmitted Jul 3, 2026• Announced Jul 6, 2026• 0 citations
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Abstract

We formulate false-vacuum decay in a mixed-field Ising chain with $1/r^α$ interactions as a spatially nonlocal Euclidean $φ^4$ theory featuring a fractional spatial kinetic term $\sim |q|^σ$, where $σ=α-1$. The nonlocal bounce is anisotropic in space-time and develops algebraic spatial tails, challenging the standard thin-wall picture of a compact droplet. Combining thin-wall arguments with numerical solutions of the full nonlocal saddle, we show that these tails preserve the leading thin-wall exponents, manifesting instead in subleading corrections. For $0<σ<1$, the lifetime exponent scales with the energy bias $h$ of the metastable state as $B\sim h^{-1/σ}$; for $1<σ<2$, the leading Coleman scaling $B\sim h^{-1}$ is recovered, while long-range effects are retained in the subdominant term $\sim h^{σ-2}$. Our results show that tunable long-range interactions fundamentally reshape bubble nucleation and alter false-vacuum decay in quantum simulators.

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We formulate false-vacuum decay in a mixed-field Ising chain with 1/rα1/r^α1/rα interactions as a spatially nonlocal Euclidean φ4φ^4φ4 theory featuring a fractional spatial kinetic term ∼∣q∣σ\sim |q|^σ∼∣q∣σ, where σ=α−1σ=α-1σ=α−1. The nonlocal bounce is anisotropic in space-time and develops algebraic spatial tails, challenging the standard thin-wall picture of a compact droplet. Combining thin-wall arguments with numerical solutions of the full nonlocal saddle, we show that these tails preserve the leading thin-wall exponents, manifesting instead in subleading corrections. For 0<σ<10<σ<10<σ<1, the lifetime exponent scales with the energy bias hhh of the metastable state as B∼h−1/σB\sim h^{-1/σ}B∼h−1/σ; for 1<σ<21<σ<21<σ<2, the leading Coleman scaling B∼h−1B\sim h^{-1}B∼h−1 is recovered, while long-range effects are retained in the subdominant term ∼hσ−2\sim h^{σ-2}∼hσ−2. Our results show that tunable long-range interactions fundamentally reshape bubble nucleation and alter false-vacuum decay in quantum simulators.
False vacuum decay describes the relaxation of a metastable state through the nucleation and growth of bubbles of the stable phase. Despite describing a broad variety of phenomena across different fields, the quantum version of the nucleation theory has little experimental or numerical support. Testing its predictions is particularly important in two or more spatial dimensions, where bubble nucleation acquires its true geometrical nature. Here, we study false vacuum decay in the quantum Ising model in two dimensions. Through tree tensor network simulations we extract the decay rate, the effective interface tension and the critical bubble size. We compare them to new semi-classical field theory calculations, and find excellent agreement. These results provide numerical evidence that the critical-bubble picture survives in an interacting quantum spin system in 2+1 dimensions.
Recent real-time simulations of false-vacuum decay in the quantum Ising chain report decay-rate prefactors that disagree with each other and with field theory by factors up to three. We prepare the metastable state by a real-time ramp of the longitudinal field, measure the survival rate per site of an infinite matrix-product state, and compare it with the decay rate that Fonseca and Zamolodchikov obtained from the free energy of the Ising field theory continued to negative field. At two lattice spacings and scaling variables y = 2.2–2.8 the ratio is 1.008 with a standard deviation of 0.015 over 6 runs, with no adjustable parameter, once the first correction to the thin-wall formula is kept. The two-kink coefficient π∆E/18 used in recent comparisons is too large by π2/9, as anticipated by Fonseca and Zamolodchikov, and exponential fits to the order parameter return rates 147–390 times larger than the nucleation rate. The survival rate, or equivalently the growth rate of the bubble density, is the observable that quantum simulators should compare with theory.
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Microscopic Dynamics of False Vacuum Decay in the 2+12+12+1D Quantum Ising Model
False vacuum decay, which is understood to happen through bubble nucleation, is a prominent phenomenon relevant to elementary particle physics and early-universe cosmology. Understanding its microscopic dynamics in higher spatial dimensions is currently a major challenge and research thrust. Recent advances in numerical techniques allow for the extraction of related signatures in tractable systems in two spatial dimensions over intermediate timescales. Here, we focus on the 2+12+12+1D quantum Ising model, where a longitudinal field is used to energetically separate the two Z2\mathbb{Z}_2Z2​ symmetry-broken ferromagnetic ground states, turning them into a ``true'' and ``false'' vacuum. Using tree tensor networks, we simulate the microscopic dynamics of a spin-down domain in a spin-up background after a homogeneous quench, with parameters chosen so that the domain corresponds to a bubble of the true vacuum in a false-vacuum background. Our study identifies how the ultimate fate of the bubble -- indefinite expansion or collapse -- depends on its geometrical features and on the microscopic parameters of the Ising Hamiltonian. We further provide a realistic quantum-simulation scheme, aimed at probing bubble dynamics on atomic Rydberg arrays.
We revisit quantum false vacuum decay for the one-dimensional Ising model, focusing on the real-time nucleation and growth of true vacuum bubbles. Via matrix product state simulations, we demonstrate that for a wide range of parameters, the full time-dependent quantum state is well described by a Gaussian ansatz in terms of domain wall operators, with the associated vacuum bubble wave function evolving according to the linearized time-dependent variational principle. The emerging picture shows three different stages of evolution: an initial nucleation of small bubbles, followed by semi-classical bubble growth, which in turn is halted by the lattice phenomenon of Bloch oscillations. Furthermore, we find that the resonant bubble only plays a significant role in a certain region of parameter-space. However, when significant, it does lead to an approximately constant decay rate during the intermediate stage. Moreover, this rate is in quantitative agreement with the analytical result of Rutkevich (Phys. Rev. B 60, 14525) for which we provide an independent derivation based on the Gaussian ansatz.